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Write the repeating decimal as a fraction.\newline.726726726.726726726

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Full solution

Q. Write the repeating decimal as a fraction.\newline.726726726.726726726
  1. Denote xx as decimal: Let's denote the repeating decimal 0.726726726...0.726726726... as xx.\newlinex=0.726726726...x = 0.726726726...\newlineTo isolate the repeating part, we multiply xx by 10001000 because there are three digits in the repeating sequence.\newline1000x=726.726726726...1000x = 726.726726726...
  2. Multiply by 10001000: Now, we subtract the original xx from 1000x1000x to get rid of the decimal part.\newline1000x−x=726.726726726...−0.726726726...1000x - x = 726.726726726... - 0.726726726...\newlineThis simplifies to:\newline999x=726999x = 726
  3. Subtract to eliminate decimal: To find the value of xx, we divide both sides of the equation by 999999.x=726999x = \frac{726}{999}
  4. Divide by 999999: We can simplify the fraction by finding the greatest common divisor (GCD) of 726726 and 999999. The GCD of 726726 and 999999 is 33. \newlinex=726/3999/3x = \frac{726 / 3}{999 / 3}\newlinex=242333x = \frac{242}{333}
  5. Simplify fraction: We check if the fraction 242333\frac{242}{333} can be simplified further. Since there are no common factors between 242242 and 333333 other than 11, the fraction is already in its simplest form.

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